Distributed Data Structures: A Survey on Informative Labeling Schemes

Distributed Data Structures: A Survey on Informative Labeling Schemes Cyril Gavoille (LaBRI, University of Bordeaux) MFCS 2006

Contents • Efficient data structures • Distributed data structures • Informative labeling schemes • Conclusion

1. Efficient data structures(Tarjan’s like) Example 1: A tree (static) T with n vertices Question: nearest common ancestor nca(x,y) for some vertices x,y? Note: queries (x,y) are not known in advance (on-line queries on a static tree)

[Harel-Tarjan ’84] Each tree with n vertices has a data structure of O(n) space (computable in linear time) such that nca queries can be answered in constant time.

Example 2: A weighted graph G with n vertices, and a parameter k≥1 Question: a k-approximation δ(x,y) on dist(x,y) in G for some vertices x,y? with dist(x,y) ≤ δ(x,y) ≤ k.dist(x,y)

[Thorup-Zwick - J.ACM ’05] Each undirected weighted graph G with n vertices, and each integer k≥1, has a data structure of O(k.n1+1/k) space (computable in O(km.n1/k) expected time) such that (2k-1)-approximated distance queries can be answered in O(k) time. Essentially optimal, related to an Erdös Conjecture.

x 2. Distributed data structures Typical questions are: Answer to query Q with the local knowledge of x (or its vicinity), so without any access to a global data structure. A network

Example 1: Distributed Hash Tables (DHT) Answer: go to w and ask it. x does not know, but w certainly knows … at least a pointer Query at x: who has any mpeg file named ‘‘Sta*Wa*’’? set of peers logical network x

Example 2: Routing in a physical network Query at x: next hop to go to y? x y

Example 3: in a dynamic setting [Afek,Awerbuch,Plokin,Saks – J.ACM ’96] It is possible to maintain a 2-approximation on the number of descendants with O(log2n) amortized messages of O(loglogn) bits each, n number of inserted vertices. Query at x: the number of descents of x (or a constant approximation of it) A growing rooted tree

Goals are: • The same as for global data structures: • Low preprocessing time • Small size data structure • Fast query time • Efficient updates + Smaller and balanced local data structures + Low communication cost (trade-offs), for multiple hops answers

3. Informative Labeling Schemes For the talk • A static network/graph • Queries: involve only vertices • Answers: do not require any communication (direct data structures)

Data Structure for graph G x y Question: dist(x,y) in a graph G? Answering to dist(x,y) consists only in inspecting the local data structure of x and of y. Main goal: minimize the maximal size of a local data structure. Wish: |DS(x,G)| « |DS(G)|, ideally |DS(x,G)| ≈ (1/n).|DS(G)|

w x y … Moreover, each vertex w  L(w) of Õ(n1/k) bits such that a (2k-1)-approximation on dist(x,y) can be answered from L(x) and L(y) only. [Thorup-Zwick - J.ACM ’05] n1+1/k Overlap: Õ(1) n1/k

Informative Labeling Schemes(more formally) [Peleg ’00] Let P be a graph property defined on pairs of vertices (can be extended to any tuple), and let F be a graph family. A P-labeling scheme for F is a pair ‹L,f› such that: GF ,u,vG: • (labeling) L(u,G) is a binary string • (decoder) f(L(u,G),L(v,G)) = P(u,v,G)

Some P-labeling schemes • Adjacency • Distance (exact or approximate) • First edge on a (near) shortest path (compact routing, labeled-based routing) • Ancestry, parent, nca, sibling relation in trees • Edge connectivity, flow • Proof labeling systems [Korman,Kutten,Peleg] • General predicate P described in monadic second order logic [Courcelle]

Ancestry in rooted trees Motivation: [Abiteboul,Kaplan,Milo ’01] The <TAG> … </TAG> structure of a huge XML data-base is a rooted tree. Some queries are ancestry relations in this tree. Use compact index for fast query XML search engine. Here the constants do matter. Saving 1 byte on each entry of the index table is important. Here n is very large, ~ 109. Ex: Is <“distributed computing”> descendant of <book_title>?

1 [22,27] 19 L(x)=[2,18] 27 8 21 20 24 23 3 7 25 10 26 9 [13,18] 4 5 6 18 12 15 11 14 17 16 Folklore? [Santoro, Khatib ’85] DFS labeling [a,b] [c,d]? 2logn bit labels

[Alstrup,Rauhe – Siam J.Comp. ’06] Upper bound: logn + O(logn) bits Lower bound: logn + (loglogn) bits 1 22 19 2 27 8 21 20 24 23 3 7 25 10 26 9 13 4 5 6 18 12 15 11 14 17 16

[Kanan,Naor,Rudich – STOC ’92] O(logn) bit labels for: • trees (and forests) • bounded arboricity graphs (planar, …) • bounded treewidth graphs Adjacency Labeling /Implicit Representation P(x,y,G)=1 iff xy in E(G) In particular: • 2logn bits for trees • 4logn bits for planar

b b f c a g e a c g c g d e e Acutally, the problem is equivalent to an old combinatorial problem: [Babai,Chung,Erdös,Graham,Spencer ’82] Small Universal Induced Graph U is an universal graph for the family F if every graph of F is isomorphic to an induced subgraph of U

b b f c a g e a c g c g d e e Universal graphU (fixed for F) Graph G of F |L(x,G)| = log2|V(U)|

Z x v y Best known results/Open questions • Bounded degree graphs: 1.867 logn [Alon,Asodi - FOCS ’02] • Trees: logn + O(log*n) [Alstrup,Rauhe - FOCS ’02]  Planar: 3logn + O(log*n) log*n = min{ i0 | log(i)n 1}

Lower bounds?: logn + (1) for planar • No hereditary family with n!2O(n) labeled graphs (trees, planar, bounded genus, bounded treewidth,…) is known to require labels of logn + (1) bits. logn + O(1) bits for this family?

y Distance P(x,y,G)=dist(x,y) in G Motivation: [Peleg ’99] If a short label (say of polylogarithmic size) can be added to the address of the destination, then routing to any destination can be done without routing tables and with a “limited” number of messages. dist(x,y) x message header=hop-count

A selection results • (n) bits for general graphs • 1.56n bits, but with O(n) time decoder! [Winkler ’83 (Squashed Cube Conjecture)] • 11n bits and O(loglogn) time decoder [Gavoille,Peleg,Pérennès,Raz ’01] • (log2n) bits for trees and bounded treewidth graphs, … [Peleg ’99, GPPR ’01] • (logn) bits and O(1) time decoder for interval, permutation graphs, … [ESA ’03]: O(n) space O(1) time data structure, even for m=(n2)

Results (cont’d) • (logn.loglogn) bits and (1+o(1))-approximation for trees and bounded treewidth graphs [GKKPP – ESA ’01] • More recently: doubling dimension- graphs Every radius-2r ball can be covered by  2 radius-r balls • Euclidean graphs have =O(1) • Include bounded growing graphs • Robust notion

Distance labeling for doubling dimension-graphs (-O() logn.loglogn) bits (1+)-approximation for doubling dimension- graphs [Gupta,Krauthgamer,Lee – FOCS ’03] [Talwar – STOC ’04] [Mendel,Har-Peled – SoCG ’05] [Slivkins - PODC ’05]

Distance labeling for planar • O(log2n) bits for 3-approximation [Gupta,Kumar,Rastogi – Siam J.Comp ’05] • O(-1log2n) bits for (1+)-approximation [Thorup – J.ACM ’04] • (n1/3)  ?  Õ(n) for exact distance • O(-1log2n) bits for (1+)-approximation for graphs excluding a fixed minor (K5,K6,…) [Abraham,Gavoille – PODC ’06]

Lower bounds for planar[Gavoille,Peleg,Pérennès,Raz – SODA ’01] n=#vertices ~ k3 #critical edges ~ k2 #labels =2k  |label|> k2/2k~ n1/3

Conclusion • Labeling scheme for distributed computing is a rich concept. • Many things remain to do, specially lower bounds

Proof Labeling Systems[Korman,Kutten,Peleg – PODC ’05] S1 v1 v3 u S2 S3 S5 v2 S4 • A graph G with a state Su at each vertex u: (G,S) • A global property P (MST, 3-coloring, …) • A marker algorithm applied on (G,S) that returns a label L(u) for u • A binary decoder (checker) for u applied on N(u): fu = f(Su,L(u),L(v1)…L(vk)) ∈ {0,1} G has property P fu=1 u G hasn't prop. P w, fw=0 whatever the labels are

What is the knowledge needed for local verifications of global properties? S1 S2 S3 S5 S4

Distributed Data Structures: A Survey on Informative Labeling Schemes